For \(k=5,6\) , the best lower bounds are due to Ball and Pepe [1]} and come from the norm graph construction of Alon, Rónyai and Szabó [2]} (originally considered for the asymmetric Zarankiewicz problem regarding bipartite graphs containing no copies of \(K_{4,7}\) ), i.e.
\(\operatorname{ex}(n,K_{6,6}) \ge \operatorname{ex}(n,K_{5,5}) = \Omega (n^{7/4}).\) 
